This problem involves finding the area under the normal distribution curve between two given values, which represent exam scores on a standardized test.

Given dataThis problem involves finding the area under the normal distribution curve between two given values, which represent exam scores on a standardized test.

Given

• Mean ($\mu$) = 538
• Standard deviation ($\sigma$) = 50
• The two values on the graph are 504.5 and 588.

To find the area under the curve (the probability), we follow these steps:

1. Convert the scores to z-scores:
   The z-score is calculated using the formula: $z = \frac{X - \mu}{\sigma}$ where $X$ is the score, $\mu$ is the mean, and $\sigma$ is the standard deviation.

2. Find the cumulative probabilities for these z-scores from a standard normal distribution table or using a calculator.

3. Subtract the cumulative probabilities to find the area under the curve between the two z-scores.

Let me calculate that for you.The shaded area under the curve between the scores of 504.5 and 588 is approximately 0.590 (to three decimal places).

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:

1. What does a z-score represent in a normal distribution?
2. How does changing the standard deviation affect the shape of a normal distribution curve?
3. How can you use the z-score to compare scores from different distributions?
4. What is the total area under a standard normal curve?
5. How do you calculate probabilities for values not exactly at 1, 2, or 3 standard deviations from the mean?

Tip: The area under the normal distribution curve between two points represents the probability that a randomly chosen value from the distribution will fall within that range.